ROLLING ELEMENT BEARING FREQUENCIES FROM 1986 FOILES REPORT, ROLLING ELEMENT BEARING FREQUENCIES 1. INTRODUCTION

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1 ROLLING ELEMENT BEARING FREQUENCIES FROM 1986 FOILES REPORT, ROLLING ELEMENT BEARING FREQUENCIES WILLIAM C. FOILES 1. INTRODUCTION FIGURE 1. Rolling element bearing geometry These calculate frequencies for rolling element bearings are calculate from the geometry, Figure 1. On actual installe bearings iniviual frequencies may exist in combination with other ones. The calculations assume no slip bounary conitions with uneforme geometry. Rolling element bearings fin many uses in toay s machinery. They can be foun in motors, fans, gas turbines, pumps, an many other machines. Some of the reasons rolling element bearings are use are: low starting friction, low operating friction, ability to support loas at low even zero spee, lower sensitivity to lubrication compare to flui film Date: March

2 WILLIAM C. FOILES bearings, thus a simpler lubrication system can often be use, an the ability to support both raial an axial loas in the same bearing. When some of these factors are important, rolling element bearings may be in use. By themselves, rolling element bearings have very little amping, so whenever a machine with rolling element bearings traverses a balance resonance, large vibration can result. Also, compare to flui film bearings which generally have a long life, rolling element bearings have a limite fatigue life ue to the repeate stresses involve in their normal use. Rolling element bearings, regarless of type ball, cylinrical, spherical, tapere, or neele consist of an inner an outer race separate by the rolling elements, which are usually hel in a cage See Figure.. Mechanical flaws may evelop on any of these components. Using the basic geometry of a bearing, the funamental frequencies generate by these flaws can be etermine. For most applications, the outer race is fixe an oes not rotate. However, in some instances, just the outer race or both races rotate. For the case of the outer race fixe, Equation 1 contains a summary of the main bearing frequencies. These frequencies may, an often o, inclue sum an ifference frequencies. Often they are moulate by the spee of the equipment.. DERIVATION OF BEARING FREQUENCIES The basic geometry for rolling element bearings can be seen in Figure 1. Balls rolling elements passing over a flaw on the outer race or the inner race generate the inner an outer roller pass frequencies. These frequencies can be erive from the bearing geometry assuming there is no slippage or imensional change with loa. Rolling element flaws or a efect in the cage can generate the roller spin frequencies; a flaw in a roller can generate a X roller spin frequency by hitting both the inner an outer races with each roller rotation. Any of a combination of flaws can generate or be moulate by the cage frequency, the frequency with which the rollers revolve as a set. What follows is a erivation of the primary rolling element bearing frequencies when both races are allowe to rotate an the simplifie form when the outer race oes not rotate.

3 ROLLING ELEMENT BEARING FREQUENCIESFROM 1986 FOILES REPORT, ROLLING ELEMENT BEARING FREQUENCIES3 FIGURE. Rolling element bearing geometry N o N i n α D N c N bpo N bpi N r outer race frequency angular spee inner race frequency angular spee number of rotating elements angular contact angle bearing pitch iameter rolling element iameter cage frequency frequency of a rolling element passing a point on the outer race frequency of a rolling element passing a point on the inner race frequency that a rolling element rotates or spins The following fact is use in the erivation: Linear velocity = angular velocity raius. Since the outer race raius, V o, is D/ + /cosα, the velocity at a point on the outer race is shown below. or V o = N o D + cosα V o = N o D + cosα Similarly, the velocity of point on the inner race is given below. V i = N i D cosα The velocity of the center of a rolling,v c, element is 1 V c = V o +V i

4 4 WILLIAM C. FOILES Thus, V c = N o D + cosα + Ni D cosα 4.1. Cage Frequency. The angular frequency of the cage is N c = V c D/ For the case where both races allowe to rotate incluing the potential for the shafts to rotate in opposite irections by using ifferent signs for N o an N i the result for the Cage Frequency is N c = N o 1 + D cosα + N i 1 D cosα 3 With the outer race fixe an the inner race with a positive rotation as is often the case: N c = N i 1 D cosα.. Outer race roller pass frequency. The frequency of a single rolling element passing a point on the outer race is the ifference in rotative spees between the outer race an the cage, N o/c. 4 N o/c = N o N c = N o 1 N o 1 + /Dcosα + Ni 1 /Dcosα This can be simplifie to the following: N o/c = N i N o 1 /Dcosα 5 The outer race ball or roller pass frequency, N bpo is the number of rolling elements times the N o/c. N bpo = n N i N o 1 /Dcosα With the outer race fixe this becomes the following. N bpo = nn i 1 /Dcosα 6 7

5 ROLLING ELEMENT BEARING FREQUENCIESFROM 1986 FOILES REPORT, ROLLING ELEMENT BEARING FREQUENCIES5.3. Inner race roller pass frequency. Similarly, the frequency of rollers passing the inner race, N bpi, is nn i/c = nn i N c or N bpi = n N i N o 1 + /Dcosα With the outer race fixe, this reuces to: 8 N bpi = nn i 1 + /Dcosα.4. Roller spin frequency. The roller spin frequency, N r, is the angular velocity of an iniviual rolling element. Equating velocities on the outer race with those of a contacting point on a rolling element, one obtains the following: V o = N o D + cosα an the velocity of a rolling element at the outer race, V o = 1 N r + N c D + cosα Equating the two, N o D + cosα = Nr + N c D + cosα 9 Solve for N r, the roller spin frequency as a positive quantity. Roller Spin Frequency with both races allowe to rotate: N r = N o N c D + cosα or in terms of N o an N i N r = N o N i D cos α D For the outer race fixe this can be written as below. 10 N r = D N i 1 D cos α 11

6 6 WILLIAM C. FOILES 3. ROLLING ELEMENT BEARING FREQUENCIES WITH FIXED OUTER RACE The funamental rolling element bearing frequencies have just been erive with both races allowe to rotate. Equations 1 present these frequencies with the outer race fixe, not rotating; this is most often the situation. In aition to these calculate rolling element frequencies, the bearing natural frequency can be generate. This frequency can be etermine best by testing, not calculation. Knowing these frequencies gives both measurement an iagnostic information. The expecte frequency ranges inform us as to the proper measurement system, an they enable us to isolate bearing-relate frequencies for iagnostics. N c = N i 1 D cosα N bpo = nn i 1 D cosα N bpi = nn i 1 + D cosα N r = D N i 1 D cos α Cage Frequency Outer Race Frequency Inner Race Frequency Roller Spin Frequency 1